I'm not a good player. I started at 30, now 36. I have a strat and a really nice Ibanez Talman acoustic, which I'm kind of in love with at the moment. When I first got it, it had 12s on it. Martins I think. I played it a lot for a month or so, then put 10s on it, which was awesome.

The other day I had this idea that if i put 13s on it, my hands would get stronger, I'd have to slow down, and I'd become a better player. So I went to the shop and got Elixir coated 13s. I was expecting more tension obviously, but I wasn't expecting my finger tips to literally slip off the strings so much. Its obviously the coating that causes this. They sound great, and they're really good when sliding but I just reckon that this slipping annoyance isn't going to go away. Maybe its something you get used to, and by getting used to something like that your accuracy on chord changes will improve too...

My intention was to play these 13s for a bit, and then switch back to 10s or even 9s. The Ernie Balls I usually use on electric say they can be used on acoustics too. Is that right? I think it would be really nice having the playability of 9s on an acoustic, but will it sound good? I guess I'll have to try and see for myself. Googling around, I can't see acoustic 9s...

One more thing... Do you eventually not even notice the difference in the board width when you switch from an acoustic to an electric, or do even pro players have to warm up for half an hour before being fluent after switching?

We generally don't use heavier-gauge strings as a training device; we use 'em because they sound better. Not that they always do, of course, but quite a lot of acoustics do have a bit of a sound improvement with more energy driving the top.
If the guitar is set up properly, there shouldn't be much difference in playing ease between lights and mediums...
Lots of people are very sensitive about neck width. Folks over at the Acoustic Guitar Forum go on and on about minor differences. Other folks (like myself) don't seem to be bothered.

Now, if I pick up my mandolin or something, a bit of adjustment is necessary.

If it was "awesome" with 10's on it then stick with them. Heavier strings may produce a tad more volume but personally I think many guitars sound better with lighter gauge strings.

You can certainly use electric strings on an acoustic. They will sound a little different to bronze but then different types of bronze sound different too - and the plain steel trebles are the same whatever sort of string you buy. I use nickel wound strings on two of my acoustic guitars: they sound fine to my ears and presumably to everyone else's since no-one has ever made an adverse comment.

And yes, if you regularly play guitars with different sized necks you soon become accustomed to changing easily - I play guitars with necks ranging from 40mm, through 43mm, 45mm, 48mm to 52mm and can switch virtually without noticing.

PS. I believe that D'Addario "Great American Bronze" have a 9 gauge option.